I'd say holding Nashville was more important.
In strategy to that part of the war, yes. Regardless you cannot downplay the effort the yankees put into the far western theater as it is perfectly comparable in size, and what was at the time importance, to the more famous operations conducted elsewhere. Lincoln knew that taking the far west would cripple southern commerce entirely because that is where they still had cotton, where they were war producing supplies, and where they were still shipping in and out of the ports. But Lincoln never could take it and failed repeatedly while trying. So instead he turned to the total war strategy of destruction exercised by Sherman and others.
So there was an Army of the Potomac west of the Mississippi?
Like Jefferson Davis, you are ignoring one of the more important tenets espoused by Sun Tzu and Nathan Bedbug Forrest among others, -- He who defends all, defends nothing.
Walt